Commensurability and representation equivalent arithmetic lattices

TL;DR

This paper shows that using representation equivalence instead of weak commensurability allows classification of arithmetic lattices without Schanuel's conjecture, and introduces a new relation called characteristic equivalence.

Contribution

It introduces the concept of representation equivalence to avoid Schanuel's conjecture and defines characteristic equivalence, simplifying classification of arithmetic lattices.

Findings

Representation equivalence replaces weak commensurability in classification.

Characteristic equivalence is a stronger relation that simplifies proofs.

Results extend to S-arithmetic settings.

Abstract

Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric spaces isospectral with respect to the Laplacian on functions. For this, in higher ranks, they assume the validity of Schanuel's conjecture. In this note, we observe that if we use the stronger notion of representation equivalence of lattices, then Schanuel's conjecture can be avoided. Further, the results are also applicable in a $S$-arithmetic setting. We also introduce a new relation on the class of arithmetic lattices, stronger than weak commensurability, which we call as characteristic equivalence, and show that it simplifies some of the arguments used in Prasad and Rapinchuk (2009) to deduce commensurability type results from weak commensurability.